What is the integral from 1 to e^9 of (6ln(x))/x ?

The integral from 1 to e^9 of (6ln(x))/x is 243

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integral from 1 to e^9 of (6ln(x))/x

\int_{1}^{e^{9}} \frac{6 ln ( x )}{x} d x

243

Solution steps

\int_{1}^{e^{9}} \frac{6 ln ( x )}{x} d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 6 \cdot \int_{1}^{e^{9}} \frac{ln ( x )}{x} d x

Apply u-substitution

: \int_{0}^{9} u d u

= 6 \cdot \int_{0}^{9} u d u

Apply the Power Rule

: [\frac{u ^{2}}{2}]_{0}^{9}

= 6 [\frac{u ^{2}}{2}]_{0}^{9}

Compute the boundaries:

\frac{81}{2}

= 6 \cdot \frac{81}{2}

Simplify

= 243

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What is the integral from 1 to e^9 of (6ln(x))/x ?
The integral from 1 to e^9 of (6ln(x))/x is 243